Binary Class AUC Optimization. As a motivating early study, [1]} points out that maximizing AUC should not be replaced with minimizing the error rate, which shows the necessity to study direct AUC optimization methods. After that, a series of algorithms have been designed for optimizing AUC. At the early stage, the majority of studies focus on a full-batch off-line setting. [2]}, [3]} optimize AUC based on a logistic surrogate loss function and ordinary gradient descent method. RankBoost [4]} provides an efficient ensemble-based AUC learning method based on a ranking extension of the AdaBoost algorithm. [5]}, [6]} constructs \(SVM^{struct}\) -based frameworks that optimize a direct upper bound of the \(0-1\)  loss version AUC metric instead of its surrogates. Later on, to accommodate big data analysis, researchers start to explore online extensions of AUC optimization methods. [7]} provides an early trial for this direction based on the reservoir sampling technique. [8]} provides a completely one-pass AUC optimization method for streaming data based on the squared surrogate loss. [9]} reformulates the squared-loss-based stochastic AUC maximization problem as a stochastic saddle point problem. The new saddle point problem's objective function only involves summations of instance-wise loss terms, which significantly reduces the burden from the pairwise formulation. [10]}, [11]} further accelerate this framework with tighter convergence rates. On top of the reformulation framework, [12]} also provides an acceleration framework for general loss functions where the loss functions are approximated by the Bernstein polynomials. [13]} proposes a novel
large-scale nonlinear AUC maximization method based on the triply stochastic gradient descent algorithm. [14]} proposes a scalable and efficient adaptive doubly stochastic gradient algorithm for generalized regularized pairwise learning problems. Beyond optimization methods, a substantial amount of researches also provide theoretical support for this learning framework from different dimensions, including generalization analysis [15]}, [16]}, [17]}, [18]}, [19]} and consistency analysis [20]}, [21]}. Last but not least, there are also some studies focusing on optimizing the partial area under the ROC curve [22]}, [23]}, [24]}. This paper takes a further step by providing an early study of the theory and algorithms for AUC-guided machine learning under the more complicated multiclass scenario.
